DOCSLIB.ORG

Bekenstein Bound of Information Number N and Its Relation to Cosmological Parameters in a Universe with and Without Cosmological Constant

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Bekenstein Bound of Information Number N and Its Relation to Cosmological Parameters in a Universe with and Without Cosmological Constant Wilfrid Laurier University Scholars Commons @ Laurier Physics and Computer Science Faculty Publications Physics and Computer Science 2013 Bekenstein Bound of Information Number N and its Relation to Cosmological Parameters in a Universe with and without Cosmological Constant Ioannis Haranas Wilfrid Laurier University, [email protected] Ioannis Gkigkitzis East Carolina University, [email protected] Follow this and additional works at: https://scholars.wlu.ca/phys_faculty Part of the Mathematics Commons, and the Physics Commons Recommended Citation Haranas, I., Gkigkitzis, I. Bekenstein Bound of Information Number N and its Relation to Cosmological Parameters in a Universe with and without Cosmological Constant. Modern Physics Letters A 28:19 (2013). DOI: 10.1142/S0217732313500776 This Article is brought to you for free and open access by the Physics and Computer Science at Scholars Commons @ Laurier. It has been accepted for inclusion in Physics and Computer Science Faculty Publications by an authorized administrator of Scholars Commons @ Laurier. For more information, please contact [email protected]. Bekestein Bound of Information Number N and its Relation to Cosmological Parameters in a Universe with and without Cosmological Constant 1 2 Ioannis Haranas Ioannis Gkigkitzis 1 Department of Physics and Astronomy, York University 4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada E-mail:[email protected] 1 Departments of Mathematics, East Carolina University 124 Austin Building, East Fifth Street, Greenville NC 27858-4353, USA E-mail: [email protected] Abstract Bekenstein has obtained is an upper limit on the entropy S, and from that, an information number bound N is deduced. In other words, this is the information contained within a given finite region of space that includes a finite amount of energy. Similarly, this can be thought as the maximum amount of information required to perfectly describe a given physical system down to its quantum level. If the energy and the region of space are finite then the number of information N required in describing the physical system is also finite. In this short letter two information number bounds are derived and compared for two types of universe. First, a universe without a cosmological constant lamda and second a universe with a cosmological constant lamda are investigated. This is achieved with the derivation of two different relations that connect the Hubble constant and cosmological constants to the number of information N. We find that the number of information N involved in a the two universes are identical or N N , and 2 2 that the total mass of the universe scales as the square root of the information number N, containing. an information number N of the order of 10122. Finally, we expressed Calogero’s quantization action as a function of the number of information N. We also have found that in self-gravitating systems the number of information N in nats is the ratio of the total kinetic to total thermal energy of the system. Key words: Bekenstein bound, cosmological constant, information, nats, entropy, mass of the universe, self-gravitating systems, Calogero's conjecture. 1. Introduction There is an upper bound for the ratio of the entropy S, to the energy E of a mass M, for any bounded system with an effective radius R. This is known as the Bekenstein (1981) upper bound given by the relation: k RE S 2 B . (1) c Where is Planck’s constant, and c the speed of light. Using Eq.(1) and the equation given in Haranas and Gkigkitzis (2013), for the entropy S to be (Lloyd, 2001): S k B N ln 2 , (2) __________________________________ In this paper both authors have contributed equally. where kB is the Boltzmann constant Eqs. (1) and (2) we obtain that the number of information N in nats contained in the quantum states in the sphere is given by the equation: 2 RE N . (3) ln 2 c Where the number of information N is given in nats. Taking, R c / H 0 to be the Hubble radius and assuming that the universe self gravitational energy E gr H 0 (Johri, 1996), we find that the number of information in nats N is equal to: 2 E 2 E 2 2 R gr N u , (4) ln 2 H ln 2 E ln 2 H ln 2 2 0 gr 0 c u Similarly Eq. (4) can be written as the ratio of the angular frequency of oscillation of the universe over its Hubble constant H, or the ratio of gravitational radius of the universe over the universe’s Compton wavelength. Next we find the entropy at the Hubble horizon to be: 2 3 5 k B c k B c c S A k . (5) H H 2 B 4G G H 0 H 0 P With reference to Haranas and Gkigkitzis (2013) and equating Eq. (2) to (5) we obtain Hubble’s parameter as a function of the number of information N to be: c H 0 , (6) ln 2 P N Substituting Eq. (6) in (4) and given that N is positive we obtain that: 2 2 2 4 E P N N . (7) 2 2 ln 2 c Solving Eq. (7) for N we obtain that: 2 2 2 2 4 E P 4 E 4 M U 0 N , (8) 2 2 ln 2 c ln 2 E P ln 2 m P c 5 c G where, and E P is the Planck energy, m P is the Planck mass, and P is the Planck G G c 3 length. Therefore we find that the number of information N at the horizon of the Friedman universe is just the square of ratio of the total energy of the universe that is attributed to the universe’s total mass to that of the Planck energy, or equivalently to the square of the total mass of the universe to that of the Planck mass. Similarly, in a universe with zero curvature and cosmological constant and with reference to Haranas and Gkigkitzis (2013) we have that the cosmological constant can be written as a function of the information number bit N in the following way: 3 3 max , (9) N ln 2 2 ln 2 N p 2 3 where max 1 / p c / G (Haranas, 2002) the maximum cosmological constant, N the number of information in nats, p is the Planck length. Therefore writing the cosmological de-Sitter horizon as function the information N we obtain: 3 N ln 2 rH P . (10) 1, 2 Substituting Eq. (10) into first part of Eq. (4) after simplifying in a way similar to Eq. (6) we obtain that: 2 2 2 4 E P N N , (11) 2 2 ln 2 c from which we obtain that: 2 2 2 2 4 E P 4 E 4 M U 0 N , (12) 2 2 ln 2 c ln 2 E P ln 2 m P and therefore we obtain that: N N , (13) 2 2 in other words the number of information contained in a Friedmann universe without cosmological constant results to a Bekenstein information bit bound that is identical to the universe involving a cosmological constant , or symbolically N N N . This is the number of information in nats with 2 F 2 the help of which information can be decompressed through matter and energy, for the two types of universe. We find that for both types of universe the number of information N depends on the same fundamental parameters i.e. the energy due to mass M over the Planck energy EP or equivalently the mass of the universe M over the Planck mass mP. From Eq. (4) we see that the number of information N constitutes the “connecting entity” of various fundamental cosmological parameters. As a result from Eq. (4) we obtain the gravitational radius of the universe can be expressed in the following way: 2 E cU R . (14) grU H 0 Next, substituting for the universe’s Compton wavelength / M c , and R 2GM / c 2 and cU U gr U 2 E M U c we obtain that the mass of the universe is given by: c 3 M U . (15) GH 0 This is an equation similar to that predicted by Hoyle (1958), Narlikar (1993), Carvalho (1995), Haranas and Harney (2009) as well as (Valev, 2010). For example Valev (2010) obtains the same equation via a dimensional analysis. Having expressed H0 as a function of the number of information N in nats, we can use Eq. (6) to express the mass of the universe as a function of the number of information N in the universe without cosmological constant to be: 2 2 c P ln 2 N c P max P M U 0.470 N 0.470 N 0.470 N . (16) G G c c P Finally, using that the Planck momentum can be written as m P c / P from the last expression of Eq. (18) we can obtain that: M U 0.470 m P N . (17) Similarly, in the universe with cosmological constant we obtain that: 2 c 2 c 3 ln 2 p max P M U N 0.470 N 0.470 N 0.470 m P N (18) G G c c P The first tem in the RHS of Eq. (18) agrees with the Eq. (8) given in Capozziello and Funkhouser (2009) and Eq. (3.8) in Funkhouser (2008). We express this equation as a function of information N, from which the number of information N in nats related to the universe’s total mass can be calculated. Therefore 53 8 taking M U 1 10 kg, (Immerman, 2001) and m P 2.176 10 kg obtain that: 2 M U 122 0 N 4.526 0.996 10 nats.
Load more

Recommended publications
  • Upper Limit on the Thermodynamic Information Content of an Action Potential
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 4 February 2020 Upper limit on the thermodynamic information content of an action potential Sterling Street∗ Department of Cellular Biology, Franklin College of Arts and Sciences, University of Georgia, Athens, GA, USA February 2, 2020 Abstract In computational neuroscience, spiking neurons are often analyzed as computing devices that register bits of information, with each ac- tion potential carrying at most one bit of Shannon entropy. Here, I question this interpretation by using Landauer's principle to estimate an upper limit for the quantity of thermodynamic information that can be dissipated by a single action potential in a typical mammalian neuron. I show that an action potential in a typical mammalian cor- tical pyramidal cell can carry up to approximately 3.4 · 1011 natural units of thermodynamic information, or about 4.9 · 1011 bits of Shan- non entropy. This result suggests that an action potential can process much more information than a single bit of Shannon entropy. 1 Introduction \The fundamental constraint on brain design emerges from a law of physics. This law governs the costs of capturing, sending, and storing information. This law, embodied in a family of equa- tions developed by Claude Shannon, applies equally to a telephone line and a neural cable, equally to a silicon circuit and a neural circuit." { Peter Sterling and Simon Laughlin [1] ∗[email protected]. Keywords: information, entropy, energy, thermodynamics, Landauer's principle, action potential 1 © 2020 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 4 February 2020 Many hypotheses and theories analyze spiking neurons as binary electro- chemical switches [1-3].
    [Show full text]
  • Edward Witten (Slides)
    Introduction to Black Hole Thermodynamics Edward Witten, IAS Madan Lal Mehta Lecture, April 19, 2021 In the first half of this talk, I will explain some key points of black hole thermodynamics as it was developed in the 1970's. In the second half, I will explain some more contemporary results, though I will not be able to bring the story really up to date. I cannot explain all the high points in either half of the talk, unfortunately, as time will not allow. Black hole thermodynamics started with the work of Jacob Bekenstein (1972) who, inspired by questions from his advisor John Wheeler, asked what the Second Law of Thermodynamics means in the presence of a black hole. The Second Law says that, for an ordinary system, the \entropy" can only increase. However, if we toss a cup of tea into a black hole, the entropy seems to disappear. Bekenstein wanted to \generalize" the concept of entropy so that the Second Law would hold even in the presence of a black hole. For this, he wanted to assign an entropy to the black hole. What property of a black hole can only increase? It is not true that the black hole mass always increases. A rotating black hole, for instance, can lose mass as its rotation slows down. But there is a quantity that always increases: Stephen Hawking had just proved the \area theorem," which says that the area of the horizon of a black hole can only increase. So it was fairly natural for Bekenstein to propose that the entropy of a black hole should be a multiple of the horizon area.
    [Show full text]
  • Unitarity Entropy Bound: Solitons and Instantons
    Unitarity Entropy Bound: Solitons and Instantons Gia Dvali Arnold Sommerfeld Center, Ludwig-Maximilians-Universit¨at, Theresienstraße 37, 80333 M¨unchen, Germany, Max-Planck-Institut f¨ur Physik, F¨ohringer Ring 6, 80805 M¨unchen, Germany and Center for Cosmology and Particle Physics, Department of Physics, New York University, 726 Broadway, New York, NY 10003, USA (Dated: July 18, 2019) We show that non-perturbative entities such as solitons and instantons saturate bounds on entropy when the theory saturates unitarity. Simultaneously, the entropy becomes equal to the area of the soliton/instanton. This is strikingly similar to black hole entropy despite absence of gravity. We explain why this similarity is not an accident. We present a formulation that allows to apply the entropy bound to instantons. The new formulation also eliminates apparent violations of the Bekenstein entropy bound by some otherwise-consistent unitary systems. We observe that in QCD, an isolated instanton of fixed size and position violates the entropy bound for strong ’t Hooft coupling. At critical ’t Hooft coupling the instanton entropy is equal to its area. I. UNITARITY AND ENTROPY The explanation given in [4] to the above phenomenon is that the key role both in saturation of the entropy The Bekenstein entropy bound [1] (see also, [2]) bound as well as in its area-form is played by an inter- particle coupling constant α that controls perturbative Smax = MR, (1) unitarity of the theory. The idea is that the objects that puts an upper limit on the amount of information stored share the property (3) are maximally packed in the sense in a system of energy M and size R.
    [Show full text]
  • Giuseppe Gaeta – List of Publications∗
    Giuseppe Gaeta { List of publications∗ Books [B1 ] G. Gaeta: \Nonlinear symmetries and nonlinear equations" (series: Math- ematics and Its Application, vol. 299); Kluwer Academic Publishers (Dor- drecht) 1994; ISBN 0-7923-3048-X [B2 ] G. Gaeta and G. Cicogna: \Symmetry and perturbation theory in non- linear dynamics" (series: Lecture Notes in Physics, vol. M57); Springer (Berlin) 1999; ISBN 3-540-65904 Monographs [M1 ] G. Gaeta: \Bifurcation and symmetry breaking"; Physics Reports 189 (1990), n. 1, 1-87 [M2 ] G. Gaeta, C. Reiss, M. Peyrard and T. Dauxois: \Simple models of DNA nonlinear dynamics"; Rivista del Nuovo Cimento 17 (1994), n.4, 1-48 Edited volumes [E1 ] D. Bambusi and G. Gaeta (eds.): \Symmetry and Perturbation Theory" (Proceedings of Torino Workshop, I.S.I., December 1996); G.N.F.M. { C.N.R. (Gruppo Nazionale di Fisica Matematica { Consiglio Nazionale delle Ricerche), Roma, 1997 [E2 ] A. Degasperis and G. Gaeta (eds.): \Symmetry and Perturbation Theory II { SPT98" (Proceedings of Roma Workshop, Universit´a\La Sapienza", December 1998); World Scientific, Singapore, 1999; ISBN 981-02-4166-6 ∗Only research and review papers and monographs are listed; in particular, this list does not include contributions to conferences or schools proceedings (as these reproduce results obtained in research papers). Note that books and monographs (not edited volumes nor textbook) also appear in the list of published research works. Last modified 5/12/2015. 1 [E3 ] D. Bambusi, G. Gaeta and M. Cadoni (eds.): \Symmetry and Pertur- bation Theory { SPT2001" (Proceedings of the international conference SPT2001, Cala Gonone, Sardinia, Italy, 6-13 May 2001); World Scientific, Singapore, 2001; ISBN 981-02-4793-1 [E4 ] G.
    [Show full text]
  • The Bekenstein Bound and Non-Perturbative Quantum Gravity
    enstein b ound and View metadata, citation and similar papers at core.ac.uk The Bek brought to you by CORE provided by CERN Document Server non-p erturbative quantum gravity Kirill V.Krasnov Bogolyubov Institute for Theoretical Physics, Kiev 143, Ukraine (March 19, 1996) We study quantum gravitational degrees of freedom of an arbitrarily chosen spatial region in the framework of lo op quantum gravity . The main question concerned is if the dimension of the Hilb ert space of states corresp onding to the same b oundary area satis es the Bekenstein b ound. We show that the dimension of such space formed by all kinematical quantum states, i.e. with the Hamiltonian constraint b eing not imp osed, is in nite. We make an assumption ab out the Hamiltonian's action, which is equivalent to the holographic hyp othesis, an which lead us to physical states (solutions of the Hamiltonian constraint) sp eci ed by data on the b oundary. The physical states which corresp ond to the same b oundary area A form a nite dimensional Hilb ert space and we show that its dimension 2 , where lies in the interval 0:55 < < 1:01. We discuss in general the is equial to exp A=l p arising notion of statistical mechanical entropy.For our particular case, when the only parameter sp ecifying the system's macrostate is the b oundary area, this entropy is given by S (A)= A.We discuss the p ossible implications of the result obtained for the black hole entropy. Among the results of the lo op approach to non-p erturbative quantum gravity the are several which, if taken seriously, tell us that the picture of geometry on scales small comparatively with our usual scales (on Planckian scales) lo oks quite di erently from the habitual picture of Riemannian geometry, and that the usual classical picture arises only on a coarse-graining.
    [Show full text]
  • Birds and Frogs Equation
    Notices of the American Mathematical Society ISSN 0002-9920 ABCD springer.com New and Noteworthy from Springer Quadratic Diophantine Multiscale Principles of Equations Finite Harmonic of the American Mathematical Society T. Andreescu, University of Texas at Element Analysis February 2009 Volume 56, Number 2 Dallas, Richardson, TX, USA; D. Andrica, Methods A. Deitmar, University Cluj-Napoca, Romania Theory and University of This text treats the classical theory of Applications Tübingen, quadratic diophantine equations and Germany; guides readers through the last two Y. Efendiev, Texas S. Echterhoff, decades of computational techniques A & M University, University of and progress in the area. The presenta- College Station, Texas, USA; T. Y. Hou, Münster, Germany California Institute of Technology, tion features two basic methods to This gently-paced book includes a full Pasadena, CA, USA investigate and motivate the study of proof of Pontryagin Duality and the quadratic diophantine equations: the This text on the main concepts and Plancherel Theorem. The authors theories of continued fractions and recent advances in multiscale finite emphasize Banach algebras as the quadratic fields. It also discusses Pell’s element methods is written for a broad cleanest way to get many fundamental Birds and Frogs equation. audience. Each chapter contains a results in harmonic analysis. simple introduction, a description of page 212 2009. Approx. 250 p. 20 illus. (Springer proposed methods, and numerical 2009. Approx. 345 p. (Universitext) Monographs in Mathematics) Softcover examples of those methods. Softcover ISBN 978-0-387-35156-8 ISBN 978-0-387-85468-7 $49.95 approx. $59.95 2009. X, 234 p. (Surveys and Tutorials in The Strong Free Will the Applied Mathematical Sciences) Solving Softcover Theorem Introduction to Siegel the Pell Modular Forms and ISBN: 978-0-387-09495-3 $44.95 Equation page 226 Dirichlet Series Intro- M.
    [Show full text]
  • The Question of an Upper Bound on Entropy
    Bekenstein has recently proposed that there exists a universal IC/82/119 •upper 'bound for the entropy of a system of energy E and maximum length INTERHAL REPORT scale R, (Limited distribution) (1) International Atomic Energy Agency and {.in the natural units c = * = k = l) with the maximum entropy being achieved Nations Educational Scientific and Cultural Organisation by a black hole. His original argument vas based on a contradiction with the INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS second lav of black hole thermodynamics if the bound did not apply. Unruh and Wald have argued that no such contradiction arises if the argument is considered in full detail. Thome has derived an internal contradiction within Bekenstein's argument by considering a non-cubic rectangular box being lowered into a black hole, and showing that the smaller, rather than the larger, length scale is required by Bekenstein's argument in that case. Since the smaller length scale could not apply in any case (it is violated THE QUESTION OF AS UPPER BOUND ON ENTROPX • by observation) he argues that Bekenstein's bound is not valid. We contend that the essence of Bekenstein's argument is not limited Asghar Qadir •• by black hole considerations. It is merely a counting of the maximum number International Centre for Theoretical physics, Trieste, Italy. of states in a finite system having a finite energy. Let the volume of the system be V and the energy E. Then the phase space volume is certainly bounded above by E^f since the momentum is bounded above by energy. For definiteness consider a cubic box of length L.
    [Show full text]
  • The Holographic Principle for General Backgrounds 2
    SU-ITP-99-48 hep-th/9911002 The Holographic Principle for General Backgrounds Raphael Bousso Department of Physics, Stanford University, Stanford, California 94305-4060 [email protected] Abstract. We aim to establish the holographic principle as a universal law, rather than a property only of static systems and special space-times. Our covariant formalism yields an upper bound on entropy which applies to both open and closed surfaces, independently of shape or location. It reduces to the Bekenstein bound whenever the latter is expected to hold, but complements it with novel bounds when gravity dominates. In particular, it remains valid in closed FRW cosmologies and in the interior of black holes. We give an explicit construction for obtaining holographic screens in arbitrary space-times (which need not have a boundary). This may aid the search for non-perturbative definitions of quantum gravity in space-times other than AdS. Based on a talk given at Strings ’99 (Postdam, Germany, July 1999). More details, references, and examples can be found in Refs. [1, 2]. 1. Entropy Bounds, Holographic Principle, and Quantum Gravity How many degrees of freedom are there in nature, at the most fundamental level? In arXiv:hep-th/9911002v1 2 Nov 1999 other words, how much information is required to specify a physical system completely? The world is usually described in terms of quantum fields living on some curved background. A Planck scale cutoff divides space into a grid of Planck cubes, each containing a few degrees of freedom. Applied to a spatially bounded system of volume V , this reasoning would seem to imply that the number of degrees of freedom, Ndof ,‡ is of the order of the volume, in Planck units: Ndof (V ) ∼ V.
    [Show full text]
  • Bekenstein Bound from the Pauli Principle∗
    Bekenstein bound from the Pauli principle∗ G. Acquaviva,1, y A. Iorio,1, z and L. Smaldone1, x 1Faculty of Mathematics and Physics, Charles University, V Holeˇsoviˇck´ach2, 18000 Praha 8, Czech Republic. Assuming that the degrees of freedom of a black hole are finite in number and of fermionic nature, we naturally obtain, within a second-quantized toy model of the evaporation, that the Bekenstein bound is a consequence of the Pauli exclusion principle for these fundamental degrees of freedom. We show that entanglement, Bekenstein and thermodynamic entropies of the black hole all stem from the same approach, based on the entropy operator whose structure is the one typical of Takahashi and Umezawa's Thermofield Dynamics. We then evaluate the von Neumann black hole{environment entropy and noticeably obtain a Page-like evolution. We finally show that this is a consequence of a duality between our model and a quantum dissipative-like fermionic system. I. INTRODUCTION identification with the standard formulae (see [7], where the bound is rigorously defined in quantum field theory This paper moves from the results of previous research (QFT) and, e.g., the review [8]) one needs to associate ge- [1], but reversing the point of view. There, Bekenstein's ometrical concepts to the Xons. For instance, one could argument that a black hole (BH) reaches the maximal make each dof correspond to one elementary Planck cell. entropy at disposal of a physical system (i.e., that it sat- Nonetheless, in our picture we do not need the exact ex- urates the Bekenstein bound [2]), leads to two main pro- pression of the bound.
    [Show full text]
  • A Proof of the Bekenstein Bound for Any Strength of Gravity Through Holography Alessandro Pesci
    A proof of the Bekenstein bound for any strength of gravity through holography Alessandro Pesci To cite this version: Alessandro Pesci. A proof of the Bekenstein bound for any strength of gravity through hologra- phy. Classical and Quantum Gravity, IOP Publishing, 2010, 27 (16), pp.165006. 10.1088/0264- 9381/27/16/165006. hal-00616263 HAL Id: hal-00616263 https://hal.archives-ouvertes.fr/hal-00616263 Submitted on 21 Aug 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Confidential: not for distribution. Submitted to IOP Publishing for peer review 14 May 2010 AproofoftheBekensteinboundforanystrengthofgravitythroughholography Alessandro Pesci∗ INFN-Bologna, Via Irnerio 46, I-40126 Bologna, Italy The universal entropy bound of Bekenstein is considered, at any strength of the gravitational interaction. A proof of it is given, provided the considered general-relativistic spacetimes allow for ameaningfulandinequivocaldefinitionofthequantitieswhichpartecipatetothebound(suchas system’s energy and radius). This is done assuming as starting point that, for assigned statistical- mechanical local conditions, a lower-limiting scale l∗ to system’s size definitely exists, being it required by holography through its semiclassical formulation as given by the generalized covariant entropy bound. An attempt is made also to draw some possible general consequences of the l∗ assumption with regards to the proliferation of species problem and to the viscosity to entropy density ratio.
    [Show full text]
  • Bekenstein Bound, Dark Matter, Dark Energy, Information Theory, Ordinary Matter
    American Journal of Computational and Applied Mathematics 2019, 9(2): 21-25 DOI: 10.5923/j.ajcam.20190902.01 On the Possible Ratio of Dark Energy, Ordinary Energy and Energy due to Information Boris Menin Mechanical & Refrigeration Consultation Expert, Beer-Sheba, Israel Abstract Information theory has penetrated into all spheres of human activity. Despite the huge number of articles on its application, the amount of scientific and technical research based on real results used in theory and in practice is relatively small. However, energy is central to life and society, both for stars and for galaxies. In particular, the energy density can be used for a consistent and unified explanation of the vast aggregate of various complex systems observed throughout Nature. This article proposes a discussion and calculations from simple and well-known studies that the energy density calculated using the information approach is a very useful metric for determining the amount of energy caused by information in the Universe. Keywords Bekenstein bound, Dark matter, Dark energy, Information theory, Ordinary matter and quantum mechanics. The results predicted future limits, 1. Introduction even far ahead, along which a road map was drawn up for adopting a set of solid conclusions regarding the limits of The definitions and methods of information theory, as computing. well as instruments and equipment developed on its basis, Some researchers suspect that eventually the axioms of have penetrated into all spheres of human activity: from quantum reconstruction will be related to information. To get biology and social sciences to astronomy and the theory of quantum mechanics out of the rules for what is allowed with gravity.
    [Show full text]
  • On an Interpolative Schr\"{O} Dinger Equation and an Alternative
    UQ Theory: October1992 On an interpolative Schr¨odinger equation and an alternative classical limit ∗ K. R. W. Jones Physics Department, University of Queensland, St Lucia 4072, Brisbane, Australia. (Revised version 13/10/92) Abstract We introduce a simple deformed quantization prescription that interpolates the classical and quantum sectors of Weinberg’s nonlinear quantum theory. The result is a novel classical limit where ¯h is kept fixed while a dimensionless mesoscopic parameter, λ ∈ [0, 1], goes to zero. Unlike the standard classical limit, which holds good up to a certain timescale, ours is a precise limit incorporating true dynamical chaos, no dispersion, an absence of macroscopic superpositions and a complete recovery of the symplectic geometry of classical phase space. We develop the formalism, and discover that energy levels suffer a generic perturbation. Exactly, they become E(λ2¯h), where λ = 1 gives the standard prediction. Exact interpolative eigenstates can be similarly constructed. Unlike the linear case, these need no longer be orthogonal. A formal solution for the interpolative dynamics is given, and we exhibit the free particle as one exactly soluble case. Dispersion is reduced, to vanish at λ = 0. We conclude by discussing some possible empirical signatures, and explore the obstructions to a satisfactory physical interpretation. arXiv:1312.4195v1 [math-ph] 15 Dec 2013 PACS numbers: 03.65.Bz, 02.30.+g, 03.20.+i, 0.3.65.Db ∗ Author’s note: Archival version of an old preprint restored from obsolete electronic media. This was first submitted to Phys. Rev. D back in 1992 but rejected as being of “insufficient interest”.
    [Show full text]